Support Vector Machine Convert optimisation problem from argmax to argmin
So actually I managed to get hold of my lecturer to explain the argmax to argmin conversion.
Generally speaking maximising $\frac{1}{||w||}$ is identical to minimising $||w||$. As $||w||$ in $\frac{1}{||w||}$ decreases, the overall value increases, i.e. we maximise it. The reason for choosing $\frac{1}{2}||w||^2$ turns out to be a less mathematic and more practical one as our optimisation algorithms used perform better on quadratic functions and the $\frac{1}{2}$ seems to be a rather arbitrary scaling choice.
If anyone has something to add, I'd love to hear any details though!